1. **State the problem:** Simplify the expression $$\sqrt[5]{1215a^{5}b} + \sqrt[5]{3a^{5}b} - a\sqrt[5]{3b}$$. 2. **Rewrite each term using properties of radicals:** Because these are fifth roots, we can write $$\sqrt[5]{x} = x^{\frac{1}{5}}$$. Also, since $$a^{5}$$ is inside a fifth root, $$\sqrt[5]{a^{5}} = a$$. 3. **Simplify each radical:** - For the first term: $$\sqrt[5]{1215a^{5}b} = \sqrt[5]{1215} \cdot \sqrt[5]{a^{5}} \cdot \sqrt[5]{b} = \sqrt[5]{1215} \cdot a \cdot \sqrt[5]{b}$$ - For the second term: $$\sqrt[5]{3a^{5}b} = \sqrt[5]{3} \cdot \sqrt[5]{a^{5}} \cdot \sqrt[5]{b} = \sqrt[5]{3} \cdot a \cdot \sqrt[5]{b}$$ - For the third term: $$a \sqrt[5]{3b}$$ (already simplified). 4. **Calculate $$\sqrt[5]{1215}$$ and note that $$1215 = 3^{5} \times 5$$:** $$1215 = 3^{5} \times 5$$ Therefore, $$\sqrt[5]{1215} = \sqrt[5]{3^{5} \times 5} = 3 \cdot \sqrt[5]{5}$$. 5. **Substitute back and factor:** $$\sqrt[5]{1215a^{5}b} = 3a \sqrt[5]{5b}$$ The second term is: $$a \sqrt[5]{3b}$$ 6. **Re-express the original expression:** $$3a \sqrt[5]{5b} + a \sqrt[5]{3b} - a \sqrt[5]{3b}$$ 7. **Combine like terms:** Note that $$a \sqrt[5]{3b} - a \sqrt[5]{3b} = 0$$ 8. **Final simplified expression:** $$3a \sqrt[5]{5b}$$. Thus, the simplified result is $$\boxed{3a \sqrt[5]{5b}}$$.